This paper is to determine the slab schedule on parallel hot rolling lines with consideration of chain precedence constraints, which restricts the slabs belonging to a chain to be successively processed without interruption in one production line. Moreover, energy loss increases as its waiting time before being processed increases, since the slabs dynamically arrive with high temperature. The objective of the problem is to minimize the total energy loss costs. Two kinds of energy loss cost that

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... are linearly and nonlinearly dependent of its waiting time are concerned. The strongly NP-hard feature of the problem stimulates us to develop branch-and-pricing algorithm to solve it. Firstly, the model is formulated as a set partitioning model. For the strongly NP-hard sub-problem, dynamic programming based on state-space relaxation method is derived to tackle it. A branch-and-bound procedure is applied for the case that the solution obtained at the root node is not integral. The whole algorithm is implemented by C language on a Pentium IV 3.0 GHz PC. Results on 25 different problem scenarios demonstrate that the proposed algorithm can solve the instances up to 120 slabs optimally within a reasonable computation time. The gaps between optimal solution and lower bound are no more than 0.0395% for the linear case and 0.0315% for the nonlinear case, respectively. Fig. 1. The flow between continuous casting and hot rolling in the Steel-making Complex.